Publication: Quotients of values of the Dedekind Eta function
Program
KU-Authors
KU Authors
Co-Authors
Xiong, Maosheng
Zaharescu, Alexandru
Advisor
Publication Date
2008
Language
English
Type
Journal Article
Journal Title
Journal ISSN
Volume Title
Abstract
Inspired by Riemann's work on certain quotients of the Dedekind Eta function, in this paper we investigate the value distribution of quotients of values of the Dedekind Eta function in the complex plane, using the form eta(A(j)z)/eta(A(j-1)z), where A(j-1) and A(j) are matrices whose rows are the coordinates of consecutive visible lattice points in a dilation X Omega of a fixed region. in R-2, and z is a fixed complex number in the upper half plane. In particular, we show that the limiting distribution of these quotients depends heavily on the index of Farey fractions which was first introduced and studied by Hall and Shiu. The distribution of Farey fractions with respect to the value of the index dictates the universal limiting behavior of these quotients. Motivated by chains of these quotients, we show how to obtain a generalization, due to Zagier, of an important formula of Hall and Shiu on the sum of the index of Farey fractions.
Description
Source:
Mathematische Annalen
Publisher:
Springer
Keywords:
Subject
Mathematics